Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 46, 2275 - 2282
Common Fixed Point Theorem for Occasionally
Weakly Compatible Mapping in Q-Fuzzy Metric
Spaces Satisfying Integral type Inequality
1
Kamal Wadhwa and Mirza Farhan Ahmed Beg2
Abstract
This paper present some common fixed point theorem for Occasionally Weakly
Compatible mapping in Q-fuzzy metric spaces satisfying integral type inequality.
Keywords: Contractive condition of integral type, Fixed point, Occasionally
Weakly Compatible mapping, Q-fuzzy metric spaces, t-norm
1. Introduction
The concept of fuzzy sets introduced by Zadeh [12] in 1965,plays an
important role in topology and analysis.Since then,there are many author to
study the fuzzy set with application. Espectially,Kromosil and Michalek[10]
put forward a new concept of fuzzy metric spaces.George and Vermani [6]
revised the notion of fuzzy metric spaces with the help of continuous tnorm.As a result of many fixed point theorem for various forms of mapping
are obtained in fuzzy metric spaces.Dhage [5] introduced the defination of Dmetric spaces and proved many new fixed point theorem in D-metric spaces.
Recently ,Mustafa and Sims[13] presented a new definition of G-metric
space and made great contribution to the development of Dhage theory. On
the other hand,Lopez-Rodrigues and Romaguera[11] introduced the concept
of Hausdorff fuzzy metric in a more general space .
The Q-fuzzy metrics spaces is introduced by Guangpeng
Sun and kai Yang[7] which can be cosider as a Generalization of fuzzy metric
spaces. Sessa [18] improved commutativity condition in fixed point theorem
by introducing the notion of weakly commuting maps in metric space.
1 Kamal wadhwa is with Govt.Narmada P.G.College, Hoshangabad, Madhya Pradesh, India.
2.Mirza Farhan Ahmed Beg is with the Bansal College of Enginering, Mandideep, Bhopal,
Madhya Pradesh India. (Corresponding author ph.no. 09229840566)
email:beg_farhan26@yahoo.com
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K. Wadhwa and M. F. A. Beg
R. Vasuki [14] proved fixed point theorems for R-weakly commuting mapping
Pant [15,16,17] introduced the new concept of reciprocally continuous
mappings and established some common fixed point theorems. The concept of
compatible maps by [10] and weakly compatible maps by [8] in fuzzy metric
space is generalized by A.Al Thagafi and Naseer Shahzad [1] by introducing
the concept of occasionally weakly compatible mappings. Recent results on
fixed point in Q-fuzzy metric space can be viewed in[7]. In this paper we have
improved the result of Guangpeng Sun and kai Yang[7] by proving the same
theorem with more weaker condition occasionally weakly compatible
mappings in Q-fuzzy metric spaces satisfying integral type inequality.
2. Preliminary Notes
Definition:2.1[2] A binary operation *:[0,1]×[0,1]→[0,1] is a continuous tnorm if it satisfy the following condition:
( i ) is associative and commutative .
(ii) is continous function.
(iii) a 1=a for all a [0,1]
(iv) a b≤c d whenever a≤c and b≤d and a,b,c,d [0,1]
Definition 2.2[7] : A 3-tuple (X,Q, ) is called a Q-fuzzy metric space if X
is an arbitrary (non-empty) set is a continuous t -norm, and Q is a fuzzy
set on X3×(0,∞),satisfying the following conditions for each x,y,z,a X
and t ,s> 0 :
(i) Q( x,x,y,t)>0 and Q(x,x,y,t)≤ Q(x,y,z,t) for all x,y,z
X with z=y
(ii) Q ( x,y,z,t)=1 if and only if x =y = z
(iii) Q(x,y,z,t) = Q(p(x,y,z),t),(symmetry) where p is a permutation
function ,
(iv) Q(x,a,a,t) Q(a,y,z,s)≤Q(x,y,z,t+s),
(v) Q(x,y,z,. ):( 0 ,∞)→[0,1] is continuous
A Q-fuzzy metric space is said to be symmetric if Q(x,y,y,t)=Q(x,x,y,t) for
all x,y X .
Definition 2.3[6] Let (X,Q, ) be a Q-fuzzy metric space.A sequence {xn} in
X converges to x if and only if Q (xm,xn,x,t) →1 as n→∞,for each t>0. It is
called a Cauchy sequence if for each 0 <ε<1 and t>0,there exist n0 N such
that Q(xm,xn,x1)>1- ε for each l,n,m ≥ n0.
Common fixed point theorem
2277
Definition 2.4[6] The Q-fuzzy metric space is called to be complete if
also
every Cauchy sequence is convergent, the sequence {xn} in X
converges to x if and only if Q(xn,xn,x,t) →1 as n→∞, for each t >0 and
it is a Cauchy sequence if for each 0 < <1 and t>0, there exist n0 N
such that Q(xm,xn,xn) > 1- for each n,m ≥ n0.
Lemma2.5[7] : If (X,Q, ) be a Q-fuzzy metric space,then Q(x,y,z,t) is nondecreasing with respect to t for all x,y,z in X .
Proof : Proof is this is implicated in [7]
Lemma2.6 [7] : Let (X,Q, ) be a Q-fuzzy metric space.
(a) If there exists a positive number k 1 such that : Q(yn+2,yn+1,yn+1,kt)≥Q(yn+1,yn,yn, t ),t >0, n N then{yn} is a Cauchy sequence
in X.
(b) if there exists k ( 0,1) such that Q(x,y,y,kt) ≥Q(x,y,y,t) for all x,y X and
t > 0 then x= y .
Proof: By the assume lim
Q x, y, z, t =1 and the property of nondecreasing, it is easy to get the results . Definition 2.7[3]: Let X be a set, f and g selfmaps of X. A point x is
called a coincidence point of f and g iff fx=gx .We shall call w=fx=gx a point
of coincidence of f and g.
Definition 2.8[7]: Let f and g be self maps on a Q-fuzzy metric space
(X ,Q,*) .Then the mappings are said to be weakly compatible if they
commute at their coincidence point, that is, f x =gx implies that
fgx = gfx .
Definition 2.9 [7]: Let f and g be self maps on a Q-fuzzy metric space
(X,Q, *) . The pair (f ,g) is said to be compatible if lim
, gf ,
gf , =1 Whenever {xn} is a sequence inX such that
lim
=lim
=z for some z
Definition 2.10[1]: Two self maps f and g of a set X are occasionally weakly
compatible (owc) iff there is a point x in X which is coincidence point of f
and g at which f and g commute.
Al-ThagaW and Naseer [1](2008) shown that occasionally weakly is weakly
compatible but converse is not true.
Example: Let R be the usual metric space. Define S T: R R by Sx = x and
Tx = x3 for all x R. Then Sx = Tx for x = 0, 2 but ST0 = TS0 and ST2 2278
K. Wadhwa and M. F. A. Beg
TS2. S and T are occasionally weakly compatible self maps but not weakly
compatible.
Lemma 2.11 [1]: Let X be a set, f , g owc self maps of X. If f and g have
unique point of coincidence, w = f x = g x, then w is the unique common
fixed point of f and g .
3. Main Result
Theorem 3.1 : Let f,T,g and S a self mapping of a complete symmetric Qfuzzy metric space with continous t-norm satisfying the following condition
(i) The pair {f, T} and {g ,S} be owc.
(ii) For all x,y in X ,k in (0,1),t>0 such that
, , ,
, , ,
, , , . , ,
φ
φ
+
where φ(t) :R R is a Lebesgue-integrable mapping which is summable,
nonegative and such that φ t 0 for each
0 then there exist a
unique point w X such that f w =Tw=w and a unique point z X such that
g z = S z =z, Moreover, z = w so that there is a unique common fixed point
of f , g ,S and T.
Proof:Let the pair {f,T} and {g ,S} be owc, so there are point x ,y X such
that fx =Tx and gy = Sy. We claim that f x= gy. If not by inequality (ii)
, , ,
, , ,
, ,
. , ,
φ
φ
,
,
, ,
,
, ,
,
,
φ
φ
Therefore f x = g y i.e. fx=Tx=gy=Sy.
Suppose that there is another point z such that f z=T z then by (1) we have
f z = T z =g y = S y , So f x = f z and w = f x = T x is the unique point of
coincidence of f and g by Lemma 2.11 w is the only common fixed point of f
and g.Similarly there is a unique point z
. such that z=gz=Sz.
Assume that w
.We have
, , , , , ,
φ
φ
,
,
,
,
, , ,
, , ,
, , ,
,
, , , φ
, , ,
, .
, , ,
φ
,
φ
,
φ
Common fixed point theorem
2279
Therefore we have z = w and by Lemma 2.11 z is unique common fixed
point of f ,g, S and T. The uniqueness of the fixed point holds from (ii).
Theorem 3.2 : Let f,T,g and S a self mapping of a complete symmetric
Q-fuzzy metric space with continous t-norm satisfying the following condition
(i) The pair {f, T} and {g ,S} be owc.
(ii) For all x,y in X ,k in (0,1), t>0 such that
,
,
,
,
,
, ,
,
φ
,
, ,
.
,
,
φ
where φ(t) :R+ R is a Lebesgue-integrable mapping which is summable,
0 for each
0 and : 0,1
0,1
nonegative and such that φ t such that
for all 0 t 1,then there exist a unique common fixed
point of f ,g ,S and T.
Proof: The proof follows from Theorem 3.1
Theorem 3.3 : Let f,T,g and S a self mapping of a complete symmetric Qfuzzy metric space with continous t-norm satisfying the following condition
(i) The pair {f, T} and {g ,S} be owc.
(ii) For all x,y in X ,k in (0,1),t>0 such that
,
,
,
,
,
, ,
,
,
, .
,
,
φ
φ
+
where φ(t) :R R is a Lebesgue-integrable mapping which is summable,
nonegative and such that φ t 0 and :[0,1]3 [0,1] such that
(t,1,t) t for all 0 t 1 then there exist a unique common fixed point of f,
g ,S and T.
Proof: Let the pair {f,T} and {g ,S} be owc, so there are point x ,y such
that f x =Tx and gy = Sy. We claim that f x= gy .If not by inequality (ii)
,
,
,
φ
,
,
,
,
,
,
,
, ,
,
,
, ,
, , ,
,
,
,
,
φ
,
,
φ
φ
Therefore f x = g y i.e. fx=Tx=gy=Sy.
Suppose that there is another point z such that f z=Tz then by (ii) we have
f z =T z =g y = S y , So f x = f z and w = f x = T x is the unique point of
coincidence of f and g by Lemma 2.11 w is the only common fixed point of f
and g.Similarly there is a unique point z . such that z=gz=Sz.
Assume that w
.We have
, , , , , ,
φ
φ
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K. Wadhwa and M. F. A. Beg
,
,
, ,
, , , ,
, , , , ,
, , ,
,
, , , ,
, , ,
,
, ,
, , ,
,
,
,
φ
φ
φ
φ
Therefore we have z = w by Lemma 2.11 and z is unique common fixed
point of f ,g, S and T. The uniqueness of the fixed point holds from (ii).
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Received: May, 2011